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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
Assertion (A): The graph $y=x^3+a x^2+b x+c$ has extremum, if $a^2<3 b$.
Reason (R): A function, $y=f(x)$ has an extremum, if $\frac{d y}{d x}>0$ or $\frac{d y}{d x}<0$ for all $x \in R$.
  • ✓
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • B
    Both (A) and (R) are true but (R) is not the correct explanation of (A).
  • C
    (A) is true but (R) is false.
  • D
    (A) is false but (R) is true.
✓

Answer

Correct option: A.
Both (A) and (R) are true and (R) is the correct explanation of (A).
(a) : For an extremum
$
\begin{aligned}
& \frac{d y}{d x}>0 \text { or } \frac{d y}{d x}<0 \text { for all } x \in R \\
\therefore & \frac{d y}{d x}=3 x^2+2 a x+b>0 \\
\Rightarrow & 3 x^2+2 a x+b>0 \Rightarrow D<0 \\
\Rightarrow & 4 a^2-4 \cdot 3 \cdot b<0 \Rightarrow a^2<3 b
\end{aligned}
$

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